#7504641

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35165	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,23)(19,25,27,22)(21,24,28,31)(26,32,29,30);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(1,4,5,8)(2,6,3,7)(10,14)(13,16)(17,19)(18,22)(20,27)(21,29)(23,25)(24,32)(26,28)(30,31);(2,3)(6,7)(10,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(18,23)(22,25)(24,31)(30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411837);